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Dept. of Math./CMA University of OsloPure Mathematics No 13ISSN 0806–2439 May 2008

On the linearization of Regge calculus

Snorre H. Christiansen∗

Abstract

We study the linearization of three dimensional Regge calculus around

Euclidean metric. We provide an explicit formula for the corresponding

quadratic form and relate it to the curlt curl operator which appears in

the quadratic part of the Einstein-Hilbert action and also in the linear

elasticity complex. We insert Regge metrics in a discrete version of this

complex, equipped with densely defined and commuting interpolators.

We show that the eigenpairs of the curlt curl operator, approximated

using the quadratic part of the Regge action on Regge metrics, converge

to their continuous counterparts, interpreting the computation as a non-

conforming finite element method.

1 Introduction

Regge calculus [49] is a combinatorial approach to Einstein’s theory of generalrelativity [61]. Space-time is represented by a simplicial complex. Given thissimplicial complex, a finite dimensional space of metrics is defined, each metricbeing determined by a choice of edge lengths. We call such metrics Regge met-rics. A functional, defined on this space of metrics and mimicking the Einstein-Hilbert action, is provided. We call this functional the Regge action. A criticalpoint of the Regge action on the space of Regge metrics is generally believed tobe a good approximation to a true solution of Einstein’s equations [42].

Regge calculus (RC) is quite popular in studies of quantum gravity [50]. Itsdiscrete nature also makes it a natural candidate for the construction of effi-cient algorithms to simulate the classical field equations, a possibility expressedalready in the last sentence of Regge’s paper. In this direction we are aware of,in particular [45][46][9][32][31]. However, it seems that the bulk of numericalrelativity computations are performed using other methods. One difficulty withsimulating Einstein’s equations is the gauge freedom (diffeomorphism invari-ance) which imposes constraints on the evolution. Hyperbolicity in this contextis a delicate matter [51]. Progress on the simulation of merging black holes [47],seems to have been achieved in large part by judiciously choosing which partialdifferential equations to solve (in particular the gauge conditions), so that anumber of traditional discretization philosophies, including finite difference, fi-nite element and collocation methods, are successfully applied today [1][34][11],in support of the emergent field of gravitational wave astronomy [56].

∗CMA, University of Oslo, PO Box 1053 Blindern, NO-0316 Oslo, Norway. email :

snorrec@math.uio.no

1

We hope that this paper can contribute to developing RC into a good al-ternative, or facilitate the integration of some of its appealing features intocurrently used methods. Its geometric “coordinate free” nature would make it astructure-preserving method in the sense of [35]. Thus our motivations are closein spirit to for instance [29][53]. The kind of variational structure that underliesRC has become a governing design principle both for finite element methodsand integration of ordinary differential equations [26][40], so that insights fromthese mature fields could well inspire decisive improvements in RC.

We are not aware of any stringent convergence results for RC, except thoseof [21]. There, it is shown that for any given smooth metric, the Regge metricsinterpolating it, have a curvature (defined by Regge calculus) which convergesin the sense of measures, when the mesh width goes to 0, to the curvature(defined in the standard way) of the given smooth metric. In numerical analysisthis would be called a consistence result. In general, consistence is only a steptowards proving convergence of a given numerical method. We also point outthat the convergence of RC is discussed, in less stringent terms, in the physicsliterature, e.g. [15] and references therein.

In [22] we related the space of Regge metrics to Whitney forms [62][63].As remarked in [14], Whitney forms correspond to lowest order mixed finiteelements [48][43], the so-called edge and face elements, for which one has a rel-atively well developed convergence theory [17][54]. More recently this analysishas been cast in the language of differential forms and related to Hodge theory[37][23][5]. We showed, in [22], that there is a natural basis for the space ofRegge metrics expressed in terms of Whitney forms and that second order dif-ferential operators restricted to Courant elements (continuous piecewise affinefunctions) are in one to one correspondence with linear forms on Regge metrics,edge elements and Courant elements. This link integrates Regge calculus intothe finite element framework. However we did not approach the question ofcurvature which is central to RC.

In this paper we further develop the theory of Regge elements. We firstinsert them in a complex of spaces equipped with densely defined interpolatorsproviding commuting diagrams as in finite element exterior calculus [4]. Thedifferential operators of this complex are those of linear elasticity. For thepurposes of relativity, it appears that less regularity is required of the fieldsthan for continuum mechanics, so that the discrete complex we obtain differsfrom those constructed in for instance [3]: the last two spaces in our complexconsist of matrix and vector valued measures that cannot be represented byintegrable functions. Next we provide results concerning the Regge action.

A priori it is not clear if RC should be considered a conforming or a non-conforming finite element method. Is the Regge action the restriction to Reggemetrics of some extension by continuity of the Einstein-Hilbert action, to alarge enough class of non-smooth metrics? One might compare with Wilson’slattice gauge theory discretization of the Yang-Mills equations [55], where thediscrete action is not a simple restriction of the continuous one. Indeed, seen asfunctionals, defined on the vector space of Lie-algebra valued (tensor product)Whitney forms, the Yang-Mills action is polynomial of order 4, whereas Wilson’saction is transcendental, even in the case of Maxwell’s equations [24].

Given a metric, the scalar curvature multiplied with the volume form pro-vided by the metric, is a certain density on space-time depending non-linearlyon the metric (and its derivatives). For a Regge metric, which has only partial

2

continuity properties between simplexes, it is not clear which, if any, of the avail-able expressions of the curvature in terms of the metric, make sense. Partialderivatives of discontinuous functions can be defined as distributions or cur-rents, in the sense of Schwartz and de Rham, but for distributions, products arenotoriously ill defined. In addition, if the background is only a piecewise affinemanifold, the associated transitions between coordinate maps are Lipschitz butnon-smooth, so that distribution theory seems inappropriate.

The arguments put forward by Regge to justify the definition of the Reggeaction are integral in nature. In RC there is a natural notion of parallel transportalong paths that avoid the codimension-2 skeleton of the simplicial complex.Around closed loops, this parallel transport behaves as if there were a curvature,concentrated only on the codimension-2 skeleton, with the expression providedby RC in terms of deficit angles. See also the justifications provided in [30].To the author, it seems desirable that this ad hoc point of view, be relatedto the contemporary mathematical theory of partial differential equations. Inthis paper we provide results concerning linearization only, but holonomies willnevertheless play a pivotal role.

If we expand, as is done for instance in [60], the Einstein-Hilbert action insmall perturbations around Minkowski space-time, the linear term is 0 since theMinkowski metric solves the Einstein equations. The first non-trivial term isa quadratic form which we denote by Q. The Euler-Lagrange equations corre-sponding to finding critical points of Q, among “all” metrics, are nothing butthe linearized Einstein equations, which describe the propagation of infinitesimalgravitational ripples on flat space-time.

It appears1 that the spatial part of Q is associated with the curlt curl oper-ator appearing in the linear elasticity complex [3], where it encodes the Saint-Venant compatibility conditions. We show that this (spatial) quadratic form,defined a priori for smooth fields, has a natural extension to (spatial) Regge met-rics. Moreover we show that this extension corresponds to the quadratic partof the Regge action, establishing that the first non-trivial terms (the secondvariations) of the Regge action and the Einstein-Hilbert action agree.

However, the natural Hilbert space on which the quadratic form is continu-ous (just) fails to contain the Regge elements! We argue that RC is a minimallynon-conforming method. As a step towards an analysis of the convergence ofnumerical methods based on RC, we show that the eigenvalues for the curlt curloperator are well approximated with Regge elements. For an alternative con-vergence result concerning linearized RC, see [10].

The theory we develop is inspired by works on the eigenvalue problem forMaxwell’s equations, [39][13][19] and also [23][4][25]. As for Maxwell’s equations,the operator does not have a compact resolvent (due to the existence of aninfinite dimensional kernel), so the basic theory [7] has to be amended. Indeed ithas been shown that in this situation, stability is not sufficient to get eigenvalueconvergence [12]. At least two additional difficulties arise. First, linked to theabove mentioned problem of hyperbolicity, is the fact that there are eigenvaluesof arbitrary magnitude of both signs. One of the signs corresponds to modesthat are excluded by the constraints in the continuous case. Due to the lack ofsign, Cea type arguments valid for Maxwell’s curl curl operator (which is positivesemi-definite), have to be replaced by inf sup conditions [6][16]. Second is the

1See the acknowledgement.

3

(limit) non-conformity of the method. Central to the argument we develop isan analogue for metrics of the Hodge decomposition of differential forms.

A number of interesting related results have been published while this paperwas under review. We mention some, that come in addition to those alreadycited. RC has been described using dual tessellations [41], in a framework rem-iniscent of the discrete exterior calculus of [28]. In a similar vein of relating RCto notions of discrete mechanics, we also point out [59], whereas [44] concerns afinite element point of view, as in [64]. Also of interest is [8]. Numerical methodsbased on differential forms have been studied on manifolds [38] and applied togeneral relativity in a simplified setting [52]. Regge elements have been redis-covered as a tool for solving equations of elasticity [27]. Hodge decompositionsof tensor fields, of the type used in this paper, have been studied in Lipschitzdomains [33].

Layout. The paper is organized as follows. In section 2 we study Regge ele-ments in finite element terms. We see what happens when we apply the Saint-Venant operator to them, and, based on the formula we obtain, insert them in adiscrete elasticity complex. In section 3 we relate the linearized Regge action tothe curlt curl operator, showing that the second variations of the Regge actionand the Einstein Hilbert action agree. In section 4 we study the discrete eigen-value problem for the Saint-Venant operator on Regge elements. An abstractframework is introduced and then applied to our case.

2 Regge elements and linear elasticity

Basis and degrees of freedom. We consider a space-slice S which is acompact flat Riemannian 3-dimensional manifold without boundary. For defi-niteness we actually consider:

S = (R/l1Z)× (R/l2Z)× (R/l3Z), (1)

for some positive reals l1, l2, l3. On S we have the Riemannian metric inheritedfrom the standard Euclidean structure of R3.

We put V = R3 and let M be the space of 3×3 real matrices. The subspace ofM consisting of symmetric matrices is denoted S, whereas that of antisymmetricmatrices is denoted A. Elements of V will be identified with 3×1 matrices calledcolumn vectors (and reals with 1 × 1 matrices). Thus the scalar product v · v′

of two vectors v, v′ ∈ V, can also be written:

v · v′ = vtv′. (2)

We let C∞(S) denote the space of smooth real functions on S. The spaceof smooth vector fields on S can be identified with C∞(S)⊗ V. We regard thegradient of a function (at a point) to be a column vector, so that we have amap:

grad : C∞(S)→ C∞(S)⊗ V. (3)

Likewise, C∞(S)⊗S can be identified with the space of smooth symmetric 3×3matrix fields.

We partition S into tetrahedrons by a simplicial complex Th. The set ofk-dimensional simplexes in Th is denoted T k

h . As is customary, the parameter

4

h denotes the largest diameter of a simplex of Th. In §4 we will be interestedin convergence results as h → 0, but until then our results concern a given hrepresenting a fixed mesh.

Regge metrics are symmetric matrix fields on S that are piecewise constantwith respect to Th and such that for any two tetrahedrons sharing a triangle asa common face, the tangential-tangential component of the metric is continuousacross the face. Thus our metrics can be degenerate – we do not impose anytriangle inequalities, as would be necessary to define a distance from the metric.The continuity property imposed on Regge metrics can also be expressed bysaying that the pullback of a metric, seen now as a bilinear form, to the interfacebetween two tetrahedrons, is the same from both sides. This space of metricshas one degree of freedom per edge, which in the non-degenerate case can betaken to be the length (or length squared) of the edge, as defined by the metric.We proceed to give basic properties of this space, including our particular choiceof basis and degrees of freedom.

In [22] we related this space of metrics, which we denote by Xh, to Whitneyforms. For each vertex x ∈ T 0

h let λx denote the corresponding barycentriccoordinate map. It is nothing but the continuous piecewise affine function takingthe value 1 at vertex x, and 0 at other vertexes. Then the following family ofmetrics is a basis of Xh, indexed by edges e ∈ T 1

h :

ρe = 1/2(

(gradλxe)(gradλye

)t + (gradλye)(gradλxe

)t)

, (4)

where the vertexes of the edge e are denoted xe and ye.We define degrees of freedom as follows. For any edge e ∈ T 1

h consider thelinear form on smooth metrics:

µe : u 7→

∫ 1

0

(ye − xe)tu(xe + s(ye − xe))(ye − xe)ds. (5)

One checks that on a tetrahedron these degrees of freedom are uni-solvent onthe constant metrics. For two edges e, e′ ∈ T 1

h we have:

µe(ρe′ ) =

{

0 if e 6= e′,1 if e = e′.

(6)

The degrees of freedom (5) make sense for some non-smooth metrics as well,in particular elements of Xh. The interpolator associated with these degreesof freedom is the projection Ih onto Xh, which to a symmetric matrix field u,associates the unique element uh ∈ Xh such that:

∀e ∈ T 1h µe(uh) = µe(u). (7)

The degrees of freedom do indeed guarantee tangential-tangential continuity ofthe interpolate.

Distributional Saint-Venant operator. Recall that the curlt curl operatoris defined on 3 × 3 matrix fields by taking first the curl of its lines, to obtaina new 3 × 3 matrix, then transposing and then taking once again the curl ofits lines. If one starts with a symmetric matrix field, the result is a symmetricmatrix field. We derive an expression for curlt curlu when u ∈ Xh. For this

5

purpose we need some expressions concerning differential operators acting ondistributions.

Let T be a Lipschitz domain in S, with outward pointing unit normal n ∈L∞(∂T )⊗ V. Let δ∂T denote the Dirac surface measure on ∂T , defined by, forφ ∈ C∞(S):

〈δ∂T , φ〉 =

∫

∂T

φ|∂T . (8)

Let u be the restriction to T , of a smooth scalar or vector field on S, whichwe extend by 0 outside T . The restriction of u to ∂T (seen from the inside ofT ) is denoted γ(u). By integration by parts in T , we have, as distributions onS:

gradu = gradT u− γ(u)n δ∂T , (9)

curlu = curlT u+ γ(u)× n δ∂T , (10)

div u = divT u− γ(u) · n δ∂T . (11)

Here, a differential operator op on the left hand side is defined in the sense ofdistributions or currents on S, whereas on the right hand side, opT denotes thecorresponding operator defined classically inside T .

Let F be a (two-dimensional) domain inside a smooth oriented hyper-surfaceof S, with piecewise smooth (one-dimensional) boundary. The oriented unitnormal on F is denoted n and the inward pointing unit normal of ∂F insidethe hyper-surface is denoted m. The Dirac surface measure on F is denoted δF ,while the double layer distribution is denoted δ′F . The Dirac line measure on∂F is denoted δ∂F . We have:

grad δF = nδ′F +mδ∂F . (12)

This formula can be applied as follows. Let u be a smooth scalar or vectorfield on S. Recall that smooth functions can be multiplied with distributions sothat the product uδF is well-defined (it is a distribution with support on F ). ALeibniz rule holds for such products. We have:

grad(uδF ) = (gradu)δF + unδ′F + umδ∂F , (13)

curl(uδF ) = (curlu)δF − u× nδ′F − u×mδ∂F , (14)

div(uδF ) = (div u)δF + u · nδ′F + u ·mδ∂F . (15)

For any 3-vector v ∈ V, let skew v ∈ A be the anti-symmetric 3 × 3 matrixdefined by:

(skew v)v′ = v × v′. (16)

Lemma 2.1. Referring to Figure 1, consider a sector in V between two half-planes F0 and F1 originating from a common edge E with unit tangent t. Theunit outward-pointing normal on the planes is denoted n. The inward pointingnormals to the edge in the planes are denoted m0 and m1. We let ni = ±nbe the normal to Fi such that (mi, ni, t) is oriented. Upon relabeling we maysuppose n0 = −n and n1 = n.

The Dirac surface measure on the boundary F = F0 ∪ F1 of the sector isdenoted δF and the double-layer distribution δ′F . The Dirac line measure on theedge is denoted δE.

6

In the sector consider a constant metric u, extended by 0 outside of it. Then:

curlt curlu =(skew n)u(skewn)δ′F+ (17)(

(skew n1)u(skewm1)− (skewn0)u(skewm0))

δE . (18)

F0

F1

m0

n0

m1n1

Et

Figure 1: View along the edge E of a sector between half-planes F0 and F1.

Proof. From formula (10) applied to each line of u we deduce:

curlu = u(skewn)δF . (19)

Transposing we get:

t curlu = −(skewn)uδF , (20)

= (skew n0)uδF0− (skewn1)uδF1

. (21)

We now apply (14) to the lines of the two matrices on the right hand side andget the result.

Remark 2.1. It is not clear from the above expression that curlt curl is a sym-metric matrix field. However if we denote by Rρ the rotation around the vectort by an angle ρ, there is an angle θ such that Rθ sends the basis (m0, n0, t) to(m1, n1, t). Put:

A(ρ) = (skewRρn0)u(skewRρm0), (22)

so that:

A(θ)−A(0) = (skewn1)u(skewm1)− (skew n0)u(skewm0). (23)

Taking derivatives with respect to ρ gives:

A′(ρ) = −(skewRρm0)u(skewRρm0) + (skewRρn0)u(skewRρn0), (24)

which is symmetric. Therefore its integral must be symmetric.

7

Proposition 2.1. Consider now a simplicial complex Th in our domain S.For any edge e ∈ T 1

h , let δe denote the Dirac line measure on e and te theunit oriented tangent vector along e. Let f be a face having e as an edge. Welet mef be the unit vector in the face f , orthogonal to the edge e and pointinginto the face. We let nef be the unit vector orthogonal to the face f orientedsuch that (mef , nef , te) is an oriented basis of V (the vector nef depends on eonly for a sign). Let [[u]]ef be the jump of u across face f in the order of nef .

We have:curlt curlu =

∑

e

[[u]]etetteδe, (25)

where we sum over edges e and put:

[[u]]e =∑

f

mtef [[u]]efnef , (26)

where we sum over faces f containing the edge e.

Proof. We use the preceding lemma. The double layer distributions cancel twoby two, by the tangential-tangential continuity of u. We have:

curlt curlu =∑

e

∑

f

(skewnef )[[u]]ef (skewmef )δe. (27)

Remark that the lines of the matrix (skewnef )[[u]]ef are proportional to nef (bytangential-tangential continuity of u). In other words there is a vector αef suchthat:

(skew nef )[[u]]ef = αefntef . (28)

Then write:

(skew nef )[[u]]ef (skewmef ) = αefntef (skewmef ) = −αef t

te. (29)

This matrix has lines proportional to te. It follows that the lines of the matrix∑

f (skewnef )[[u]]ef (skewmef ) must be proportional to te. Since in addition thismatrix is symmetric, it can be written:

∑

f

(skewnef )[[u]]ef (skewmef ) = setette . (30)

The scalar coefficient se is determined by taking traces:

se = tr∑

f

(skewnef )[[u]]ef (skewmef ), (31)

= − tr∑

f

(skew nef )[[u]]ef (tentef − nef t

te), (32)

= −∑

f

ntef (skewnef )[[u]]ef te −

∑

f

tte(skewnef )[[u]]efnef , (33)

=∑

f

mtef [[u]]efnef , (34)

as announced.

One can compare with the approach of [36], where tangential-tangentialcontinuity is not enforced on the symmetric matrix fields. The discrete curvaturethey define is based on inter-element jumps, in an expression corresponding tothe double layer distribution in (17).

8

A discrete elasticity complex. The space Xh can be inserted in a complexof spaces Xk

h with 0 ≤ k ≤ 3, each equipped with a densely defined interpolatorIkh .

(0) We let X0h denote the space of continuous piecewise affine vector fields.

It consists of the vector fields of the form:

v =∑

x∈T 0h

vxλx, (35)

for all choices of vectors vx ∈ V assigned to vertexes x ∈ T 0h . We equip X0

h withthe nodal interpolator I0h, which to any v ∈ C∞(S)⊗ V associates:

I0hv =∑

x∈T 0h

v(x)λx. (36)

(1) We put X1h = Xh, and equip it with the interpolator I1h = Ih. For any

element of X0h, its symmetrized gradient is piecewise constant and tangential-

tangential continuous. In other words, the deformation operator, denoted def,induces a map X0

h → X1h. For our choice of bases of these spaces we notice that

if e is an edge with vertexes x and y, we have for any v ∈ V:

µe(def vλx) = (y − x)tv. (37)

(2) We let X2h denote the space of matrix valued edge measures of the form:

u =∑

e∈T 1h

uetetteδe, (38)

where for each edge e, ue is a real number, te is the unit oriented tangent vectorto e and δe is the Dirac line measure on e. Proposition 2.1 shows that curlt curlinduces a map X1

h → X2h. The standard L2 duality on matrix fields extends to

a non-degenerate bilinear form on X2h ×X

1h, which we denote by 〈·, ·〉. Notice

that, comparing with the definition of degrees of freedom µe on X1h in equation

(5), we have for any u ∈ X1h:

〈tetteδe, u〉 =

∫

e

tteute, (39)

= µe(u)/le, (40)

where le is the Euclidean length of the edge e. Therefore the interpolator I1hdeduced from the degrees of freedom µe satisfies for any u ∈ C∞(S)⊗ S:

∀v ∈ X2h 〈v, I1hu〉 = 〈v, u〉. (41)

Just as the elements of X2h act as degrees of freedom for X1

h, the elementsof X1

h can be used as degrees of freedom for X2h. The associated interpolator I2h

onto X2h is determined by the property that for any u ∈ L2(S)⊗S, I2hu satisfies:

∀v ∈ X1h 〈I2hu, v〉 = 〈u, v〉. (42)

We remark that, by (41) and (42), for all u ∈ L2(S)⊗ S and all v ∈ C∞(S)⊗ S:

〈I2hu, v〉 = 〈I2hu, I

1hv〉 = 〈u, I

1hv〉. (43)

9

In other words, I1h and I2h are adjoints of one another.(3) We let X3

h denote the space of vector vertex measures of the form:

u =∑

x∈T 0h

uxδx, (44)

where for each vertex x, ux ∈ V is a vector and δx is the Dirac measure attachedto x. The elements of X3

h act as natural degrees of freedom for X0h, yielding the

nodal interpolator I0h of X0h. The standard L2 duality on vector fields extends

to a non-degenerate bilinear form on X3h × X0

h, denoted 〈·, ·〉. We define aninterpolator I3h onto X3

h by requiring, for any u ∈ L2(S)⊗ V:

∀v ∈ X0h 〈I3hu, v〉 = 〈u, v〉. (45)

As in the preceding case we remark that for all u ∈ L2(S) ⊗ V and all smoothv ∈ C∞(S)⊗ V:

〈I3hu, v〉 = 〈I3hu, I

0hv〉 = 〈u, I

0hv〉. (46)

One also checks that for an edge e with unit tangent te and vertexes x and y,such that te = (y − x)/|y − x|, we have:

div(tetteδe) = te(δy − δx). (47)

This shows that the divergence operator induces a map X2h → X3

h.

That concludes the list of spaces and operators we need to form our diagram.The following theorem summarizes some of the above remarks, and relates thecontinuous elasticity complex [3] to a discrete one.

Theorem 2.2. We have a commuting diagram of spaces:

C∞(S)⊗ Vdef

//

I0h

��

C∞(S)⊗ Scurlt curl

//

I1h

��

C∞(S)⊗ Sdiv

//

I2h

��

C∞(S)⊗ V

I3h

��

X0h

def// X1

hcurlt curl

// X2h

div// X3

h

(48)On the lower row the linear operators are defined in the sense of distributions.

Proof. Only commutativity remains to be proved.(i) For any u ∈ C∞(S)⊗V and any edge e ∈ T 1

h with vertexes x, y, we have:

µe(def u) = tte(u(y)− u(x)). (49)

We deduce :µe(def u) = µe(def I

0hu). (50)

Commutation of the first square follows.(ii) For any u ∈ C∞(S)⊗ S and any v ∈ X1

h, we have:

〈I2h curlt curlu, v〉 = 〈curlt curlu, v〉, (51)

= 〈curlt curl v, u〉, (52)

= 〈curlt curl v, I1hu〉, (53)

= 〈curlt curl I1hu, v〉, (54)

10

so that:

I2h curlt curlu = curlt curl I1hu. (55)

This proves commutation of the middle square.(iii) Commutation of the last square follows from (i) by duality.

The discrete fields considered here have less regularity than those defined in[2], which are all at least square integrable, throughout the complex. Spacesof matrix fields adapted to a second order differential operator have also beenconsidered in [58]. In their case, the differential operator is “div div” (extract-ing first the divergence, per line say, of the matrix field and then the divergenceof the obtained vector field), for which normal-normal continuity of the ma-trix fields is the natural analogue of our tangential-tangential continuity. Notealso that the two-dimensional case of the “Regge complex” was studied in [27](independently of any reference to Regge calculus).

3 Linearizing the Regge action

The setting and notations are as in the preceding section. Our aim is to computethe second variation of the Regge action and relate it to the previously exhibitedSaint-Venant operator.

We consider first what happens around a single edge. Fix an oriented line(edge) in V with unit tangent t. The set-up is similar to the one of Figure 1.Originating from this edge are half-planes (faces) indexed by a cyclic parameterf and ordered counter-clockwise. Thus the face coming immediately after f isdenoted f +1. The sector between faces f and f +1 is indexed by f +1/2. Letmf be the oriented unit length vector in the half-plane f which is orthogonalto t. Let nf be the normal to half-plane f , so that (mf , nf , t) is an orientedorthonormal basis of V.

Each sector between half-planes f and f + 1 is equipped with a constantmetric uf+1/2 = uf+1/2(ǫ) depending on a small parameter ǫ. We suppose thatu(ǫ) is continuous across the half-plane f in the tangential-tangential directions.We suppose that uf+1/2(0) is the canonical Euclidean metric on V, and we willbe particularly interested in the first derivative of uf+1/2(ǫ) with respect to ǫat ǫ = 0, which we denote by u′f+1/2.

To ease notations, the dependence upon ǫ will be implicit in what follows.The derivative of a function ψ : ǫ 7→ ψ(ǫ) at ǫ is denoted ψ′(ǫ). Unless otherwisespecified we only differentiate at ǫ = 0 and therefore write ψ′ = ψ′(0).

We want to parallel transport a vector around the edge, by a path going oncearound it and with respect to the metric u(ǫ). In each sector parallel transportis trivial, but from one sector to another, say from f − 1/2 to f +1/2 we denoteby Tf the matrix of the parallel transport in the basis (mf , nf , t). It is definedas follows. The oriented unit normal to face f with respect to uf−1/2 is denoted

k−f , that with respect to uf+1/2 is denoted k+f . The operator Tf maps the basis

(mf , k−

f , t) to (mf , k+f , t) . We denote by Rg

f the matrix of the identity operatorfrom basis (mf , nf , t) to basis (mg, ng, t).

The holonomy from sector f −1/2 to itself, in the basis (mf , nf , t) is definedto be:

Ef−1/2 = Rff−1Tf−1 · · ·R

g+1g Tg · · ·R

f+1f Tf . (56)

11

Remark that Tf is an isometry from the metric uf−1/2 to the metric uf+1/2.Consequently Ef−1/2 is an isometry with respect to uf−1/2. In the plane or-thogonal to t it is a simple rotation. The angle of this rotation does not dependon f and is the deficit angle associated with the edge t. We denote it by θ.

Proposition 3.1. The derivative of the deficit angle θ at ǫ = 0 is given as asum of jumps:

θ′ = 1/2∑

f

mtf(u

′f+1/2 − u

′f−1/2)nf . (57)

Proof. Let mf be the oriented unit normal to t in face f , with respect touf−1/2 or equivalently uf+1/2. Let Pf be the matrix of the identity from basis

(mf , nf , t) to (mf , k−

f , t). Since (mf , k−

f ) is an orthonormal oriented basis of theplane orthogonal to t, with respect to the metric induced by uf−1/2 we have:

Ef−1/2 = P−1f

cos θ − sin θ 0sin θ cos θ 0

0 0 1

Pf . (58)

Differentiating this expression at ǫ = 0, using that Pf (0) is the identity matrix,we get:

E′f−1/2 =

0 −θ′ 0

θ′ 0 0

0 0 0

. (59)

Differentiating (56) at ǫ = 0 we obtain, since Tf(0) is the identity matrix:

E′f−1/2 = Rf

f−1T′f−1R

f−1f + · · ·+Rf

gT′gR

gf + · · ·+ T ′

f . (60)

Let J be the canonical skew matrix:

J =

0 −1 01 0 00 0 0

. (61)

It commutes with all Rgf . We have:

θ′ = −1/2 tr(JE′f−1/2) (62)

= −1/2∑

g

tr(JT ′g). (63)

We determine the terms in this sum. Let M±g be the matrix in (mg, ng, t) of

the operator sending (mg, ng, t) to (mg, k±g , t). We have:

Tg =M+g (M−

g )−1. (64)

Differentiating with respect to ǫ at ǫ = 0 we obtain, since M±g is the identity

matrix at ǫ = 0:T ′g = (M+

g )′ − (M−g )′. (65)

Define reals α±g , β

±g , γ

±g by:

M±g =

1 α±g 0

0 β±g 0

0 γ±g 1

. (66)

12

Then:tr(JT ′

g) = (α+g )

′ − (α−g )

′. (67)

Differentiating the identity :

mtgug+1/2k

+g = 0, (68)

at ǫ = 0 yields:mt

gu′g+1/2ng +mt

g(k+g )

′ = 0. (69)

Since also, by the definition (66):

α+g = mt

gk+g , (70)

it follows that:(α+

g )′ = −mt

gu′g+1/2ng. (71)

Similarly we have:(α−

g )′ = −mt

gu′g−1/2ng. (72)

Insert these two expressions in (67) and combine with (63) to get the proposition.

We consider now the general case of a simplicial complex in S. For a Reggemetric u ∈ Xh its Regge action R(u) is defined as follows. For any edge e itslength, as determined by u, is denoted le(u) and its deficit angle θe(u). Then,summing over edges, one defines [49]:

R(u) =∑

e

θe(u)le(u). (73)

Proposition 3.2. Let Regge metrics u(ǫ) ∈ Xh depend smoothly on a smallreal parameter ǫ. We use the notations:

R(ǫ) = R(u(ǫ)), θe(ǫ) = θe(u(ǫ)), le(ǫ) = le(u(ǫ)). (74)

We suppose that u(0) is the constant Euclidean metric. Then we have:

R(ǫ) = ǫ2/4∑

e

θ′e(0)le(0)−1µe(u

′(0)) +O(ǫ3). (75)

Proof. We have:θe(0) = 0. (76)

Consequently:

R(ǫ) =∑

e

(ǫθ′e(0) + ǫ2/2 θ′′e (0)) (le(0) + ǫl′e(0)) + O(ǫ3) (77)

= ǫ∑

e

θ′e(0)le(0) + ǫ2∑

e

(θ′e(0)l′e(0) + 1/2 θ′′e (0)le(0)) + O(ǫ3). (78)

It is a remarkable property of Regge calculus, proved in [49], that for any ǫ:

∑

e

θ′e(ǫ)le(ǫ) = 0. (79)

13

This handles the first term in (78). Differentiating we get in addition:

∑

e

θ′′e (0)le(0) + θ′e(0)l′e(0) = 0. (80)

Inserting this in the second term of (78) we get:

R(ǫ) = ǫ2/2∑

e

θ′e(0)l′e(0) +O(ǫ

3). (81)

Finally consider an edge e and denote its extremities by x and y. We havele(0) = |y − x| and te = (y − x)/|y − x|. We can write:

le(ǫ) = ((y − x)tu(ǫ)(y − x))1/2 (82)

= (|y − x|2 + ǫ(y − x)tu′(0)(y − x) +O(ǫ2))1/2 (83)

= le(0)(1 + ǫ/2 tteu′(0)te +O(ǫ

2), (84)

so that:

l′e(0) = 1/2 le(0)tteu

′(0)te, (85)

= 1/2 le(0)−1µe(u

′(0)). (86)

This completes the proof.

Now insert expressions (57) and (40) in (75) and compare with the expres-sions (25) and (26) for the curlt curl operator. We conclude:

Theorem 3.3. With the above notations we have the expansion:

R(u(ǫ)) = ǫ2/8 〈curlt curlu′(0), u′(0)〉+O(ǫ3). (87)

It can be checked that curlt curl is also the operator appearing in the spatialpart of the second variation of the Einstein Hilbert action. It suffices to writecurlt curl in coordinates (index notation) and compare with the expression forthe second variation provided in, for instance, [60].

4 Eigenvalue approximation

An abstract setting. In this paragraph we develop a convergence theory forsome non-conforming eigenvalue approximations. It is a generalization of thepoint of view detailed in [25] for conforming approximations of semi-definiteoperators. Some of our arguments are closer to a discontinuous Galerkin wayof thinking [18] but we also need to cover the case of operators with spectrumin both ends of the real axis, so that coercivity should be replaced by inf supconditions [6].

Let O be a Hilbert space, with scalar product 〈·, ·〉. The associated norm isdenoted | · |. Let X be a dense subspace of O, which is also a Hilbert space suchthat the injection X → O is continuous. The norm of X is denoted ‖ · ‖. Wesuppose that we have a symmetric continuous bilinear form a on X . We do notrequire a to be semi-definite. We are interested in the eigenvalue problem, tofind u ∈ X and λ ∈ R such that:

∀v ∈ X a(u, v) = λ〈u, v〉. (88)

14

We define:

W = {u ∈ X : ∀v ∈ X a(u, v) = 0}, (89)

V = {u ∈ X : ∀v ∈ W 〈u, v〉 = 0}. (90)

These are closed subspaces of X and we have a direct sum decomposition:

X = V ⊕W. (91)

We suppose that W is closed in O and that the injection V → O is compact.Let V be the closure of V in O. Let P be the orthogonal projection in O withrange V and kernel W .

We suppose that the map X → X⋆, u 7→ a(u, ·) determines an isomorphismV → V ⋆, equivalently:

infu∈V

supv∈V

|a(u, v)|

‖u‖ ‖v‖> 0. (92)

The dual of X with O as pivot space is denoted X ′. Notice that we distin-guish it from the space of continuous linear forms on X which is denoted X⋆.For instance O is a dense subspace of X ′ but not of X⋆. The map u 7→ 〈u, ·〉defines an isomorphism X ′ → X⋆. The notation Y ′ will be used later, with thesame meaning, for other dense subspaces Y of O.

Define a bounded operator K : X ′ → X as follows. To any u ∈ X ′ associateKu = v ∈ V such that:

∀w ∈ V a(v, w) = 〈u,w〉. (93)

As an operator O → O, K is compact and selfadjoint. This guarantees thatthere is an orthonormal basis of O consisting of eigenvectors of K. The non-zeroeigenvalues are inverses of those of (88), with corresponding eigenspaces.

We now turn to the the approximation of the eigenproblem (88) by a Galerkinmethod.

Suppose that (Xn) is a sequence of finite-dimensional subspaces of O, andthat for each n ∈ N we have a symmetric bilinear form an on Xn. We solve theeigenvalue problems: find u ∈ Xn and λ ∈ R such that:

∀v ∈ Xn an(u, v) = λ〈u, v〉. (94)

Decompose as before:

Wn = {u ∈ Xn : ∀v ∈ Xn a(u, v) = 0}, (95)

Vn = {u ∈ Xn : ∀v ∈ Wn 〈u, v〉 = 0}. (96)

We have a direct sum decomposition:

Xn = Vn ⊕Wn. (97)

Define also Kn : O → O as follows. To any u ∈ O associate Knu = v ∈ Vn suchthat:

∀w ∈ V an(v, w) = 〈u,w〉. (98)

SinceKn has finite rank it is compact. It is also selfadjoint. Non-zero eigenvaluesof Kn are the inverses of those of (94), with corresponding eigenspaces.

15

For the eigenpairs of Kn to converge to those of K in a natural sense, thefollowing should be achieved [20]:

‖K −Kn‖O→O → 0. (99)

Our aim now is to devise sufficient conditions for this to hold. For insights intosome difficulties that arise in the context of discretizations of the form (94),when a has infinite dimensional kernel, see in particular [12].

For two real sequences (an) and (bn), estimates of the form, there existsC > 0, independent of the sequences, such that for all n:

an ≤ Cbn, (100)

will be written:

an 4 bn or bn < an. (101)

We also use the notation:

an ≈ bn, (102)

when:

an 4 bn and bn 4 an. (103)

For the applications we have in mind, Xn is not a subspace of X , but wecan weaken the norm of X to obtain a space containing Xn, without loosingessential compactness properties. Precise statements follow. We suppose thatwe have two more Hilbert spaces X− and X+ with inclusions:

X+ ⊂ X ⊂ X− ⊂ O, (104)

which are continuous and have dense range. The norms of X+ and X− aredenoted ‖ · ‖+ and ‖ · ‖− respectively. We suppose that a is continuous onX+ ×X−. We define:

W+ = {u ∈ X+ : ∀v ∈ X− a(u, v) = 0}, (105)

W− = {u ∈ X− : ∀v ∈ X+ a(u, v) = 0}, (106)

V + = {u ∈ X+ : ∀v ∈ W+ 〈u, v〉 = 0}, (107)

V − = {u ∈ X− : ∀v ∈ W− 〈u, v〉 = 0}. (108)

We have direct sum decompositions:

X+ = V + ⊕W+ and X− = V − ⊕W−. (109)

We suppose that W+ is closed in O, which yields:

W+ =W =W−. (110)

We suppose also that that we have the inf-sup conditions:

infu∈V +

supv∈V −

|a(u, v)|

‖u‖+ ‖v‖−> 0. (111)

It follows that K is bounded X−′ → X+.

16

We also suppose that the injection V − → O is compact. By duality theinjection V → X−′ is compact, so that, by composition, K is compact as anoperator O → X+.

Concerning our spaces Xn we suppose that they are equipped with projec-tions Qn : O → Xn which are uniformly bounded O → O and map W to Wn.Moreover we suppose that δ(Wn,W ) → 0, the gap being calculated with thenorm of O. Recall the definition:

δ(Wn,W ) = supu∈Wn

infv∈W|u− v|/|u|. (112)

We impose of course that for any u ∈ O there is a sequence un ∈ Xn suchthat un → u in O. We then have Qnu→ u in O.

Lemma 4.1. Consider sequences un = vn + wn with vn ∈ Vn and wn ∈ Wn.Suppose that un → u in O, and decompose u = v + w with v ∈ V and w ∈ W .Then vn → v and wn → w in O.

Proof. We have:

0← |u− un|2 = |v − vn|

2 + |w − wn|2 − 2〈v, wn〉 − 2〈vn, w〉. (113)

Considering the right hand side we have:

|〈v, wn〉| = |〈v, wn − (I − P )wn〉| ≤ |v| |wn| δ(Wn,W ) (114)

≤ |v| |un| δ(Wn,W )→ 0, (115)

and also:

|〈vn, w〉| = |〈vn, w −Qnw〉| ≤ |vn| |w −Qnw| (116)

≤ |un| |w −Qnw| → 0. (117)

This gives the lemma.

The spaces Xn are not necessarily subspaces of X , but always of X−.Since W is closed in X−, the following proposition states that, as n → ∞,δ(Vn, V

−)→ 0, the gap being calculated with the norm of X−.

Proposition 4.1. There is a sequence ǫn → 0 such that for all u ∈ Vn:

|u− Pu| ≤ ǫn‖u‖−. (118)

Proof. For any u ∈ Vn we have:

u−QnPu = Qn(u − Pu) ∈ Wn. (119)

We write:

|Pu−QnPu|2 = |Qn(u− Pu)− (u − Pu)|2 (120)

= |Qn(u− Pu)|2 + |u− Pu|2 − 2〈Qn(u− Pu), Pu〉, (121)

from which it follows that:

|u− Pu|2 ≤ |Pu−QnPu|2 + 2δ(Wn,W ) |Qn(u− Pu)| |Pu|. (122)

Recall the uniform boundedness of the Qn : O → O. The first term on the righthand side is handled by the fact that P is compact as a map X− → O. Thesecond term is handled by the gap property of Wn.

17

The forms an are required to be consistent with a in the following sense. Forall u ∈ X+ there is a sequence un ∈ Xn such that un → u in O and:

limn→∞

supv∈Xn

|a(u, v)− an(un, v)|

‖v‖−= 0. (123)

Moreover we suppose that we have the following weak inf-sup condition, uniformin n:

1 4 infu∈Vn

supv∈Vn

|an(u, v)|

|u| ‖v‖−. (124)

Proposition 4.2. Under the above circumstances we have discrete eigenpairconvergence in the sense of (99).

Proof. (i) Define Pn : X+ → Xn as follows. To any u ∈ X+ associate Pnu =v ∈ Vn such that:

∀w ∈ Vn an(v, w) = a(v, w). (125)

It follows from (124) that Pn is uniformly bounded X+ → O. To get pointwiseconvergence, pick u ∈ X+. Choose un ∈ Xn converging to u in O, from theconsistence hypothesis (123). Decompose un = vn + wn with vn ∈ Vn andwn ∈Wn. By Lemma 4.1, we have vn → Pu in O. We also have:

|Pnu− vn| 4 supw∈Vn

|an(Pnu− vn, w)|

‖w‖−= sup

w∈Vn

|a(u,w)− an(un, w)|

‖w‖−→ 0. (126)

We deduce Pnu→ Pu in O.We remark that for u ∈ V we have:

∀v ∈ Vn an(PnKu, v) = a(Ku, v) = 〈u, v〉 = an(Knu, v), (127)

so that:

PnKu = Knu. (128)

Moreover, as already pointed out, K is compact as an operator O → X+.Combining these remarks we get:

‖K −Kn‖V→O → 0. (129)

(ii) For u ∈ W we have:

∀v ∈ Vn an(Knu, v) = 〈u, v〉 = 〈u, v − Pv〉. (130)

Using (124) and Proposition 4.1, we can write:

|Knu| 4 supv∈Vn

|an(Knu, v)|

‖v‖−= sup

v∈Vn

〈u, v − Pv〉

‖v‖−(131)

4 ǫn|u|. (132)

This gives:

‖Kn‖W→O → 0. (133)

Since K is zero on W this concludes the proof.

18

Application to Regge calculus. We first define some function spaces thatwill allow us to use the preceding setting.

We define O = L2(S)⊗ S, equipped with the canonical scalar product, andlet a be the bilinear form on symmetric matrix fields, which, at least for smoothones is defined by:

a(u, v) = 〈curlt curlu, v〉. (134)

We define for any α ∈ [−1, 1]:

Xα = {u ∈ O : curlt curlu ∈ H−1+α(S)⊗ S}. (135)

Fix now α ∈]0, 1/2[ and put:

X+ = Xα, (136)

X = X0, (137)

X− = X−α. (138)

Proposition 4.3. The required hypotheses are satisfied:

• a defines a continuous bilinear form on X,

• the kernel W is closed in O,

• V is compactly embedded in O,

• a is invertible on V × V .

Moreover:

• a defines a continuous bilinear form on X+ ×X−,

• the kernel W+ is closed in O,

• V − is compactly embedded in O,

• the inf sup condition (111) holds.

Proof. We base the proof on Fourier analysis. For any ξ ∈ V we denote byF (ξ) : V→ C the associated Fourier mode:

F (ξ) : x 7→ exp(iξ · x). (139)

For any a ∈ S we have:

curlt curl(aF (ξ)) = skew(ξ)a skew(ξ)F (ξ). (140)

For ξ 6= 0 choose two normalized vectors ξ1, ξ2 ∈ V such that (ξ, ξ1, ξ2) is anoriented orthogonal basis of V. We make some remarks:

First, a ∈ S satisfies aξ = 0 iff a is a linear combination of the three matrices:

σ1(ξ) = ξ1ξt1 + ξ2ξ

t2 , (141)

σ2(ξ) = ξ1ξt2 + ξ2ξ

t1 , (142)

σ3(ξ) = ξ1ξt1 − ξ2ξ

t2 . (143)

19

We compute:

skew(ξ)σ1(ξ) skew(ξ) = −|ξ|2σ1(ξ), (144)

skew(ξ)σ2(ξ) skew(ξ) = |ξ|2σ2(ξ), (145)

skew(ξ)σ3(ξ) skew(ξ) = −|ξ|2σ3(ξ). (146)

Second, a ∈ S is orthogonal to these three matrices iff:

skew(ξ)a skew(ξ) = 0. (147)

In fact this occurs iff a can be written, for some (uniquely determined) b ∈ V:

a = bξt + ξbt. (148)

Since we restricted attention to periodic boundary conditions, L2(S)⊗ C isspanned by Fourier modes:

F (k) : k ∈ (2π/l1)Z× (2π/l2)Z× (2π/l3)Z. (149)

The preceding remarks then provide a Hilbertian basis of O consisting of eigen-vectors for curlt curl. Taking into account the characterization of Sobolevspaces in terms of Fourier series, all the claimed results follow.

We consider a quasi-uniform sequence of simplical meshes Th of S, with meshwidth h → 0. Attached to Th we denote as before by Xh = X1

h the space ofRegge metrics, and equip it with the bilinear form ah induced by a, defined foru, v ∈ X1

h by:ah(u, v) = 〈curlt curlu, v〉, (150)

with respect to the canonical duality pairing 〈·, ·〉 on X2h ×X

1h.

The elements of X2h are not in H−1(S)⊗ S since traces on edges are not well

defined in H1(S), therefore Xh is not a subspace of X . One might therefore bereluctant to call ah the restriction of a. However, consistence of ah with a is notin doubt. We remark also that Wh is a subspace of W (since X1

h acts as degreesof freedom for X2

h) so that δ(Wh,W ) = 0. We will use that traces on edges arewell defined H1+α(S)→ L2(e), from which it follows that Xh is in X−.

Proposition 4.4. The spaces Xh can be equipped with projections Qh that areuniformly bounded O→ O and map W to Wh.

Proof. We use the technique of [4], which was based on the earlier works [57][23].Let φ : R3 → R be smooth, non-negative with support in the unit ball and

integral 1. Define its scaling by:

φh(x) = h−3φ(h−1x). (151)

Let Rǫh be regularization by convolution by φǫh. Remark that it commutes with

constant coefficient differential operators. Composing with the already definedinterpolator Ih we get an operator IhR

ǫh : O → Xh. We can fix a small ǫ such

that for all h:‖(id− IhRh)|Xh

‖O→O ≤ 1/2. (152)

We may then put:Qh = (IhRh|Xh

)−1IhRh, (153)

to get the required projection.

20

Proposition 4.5. We have an estimate, uniform in h:

∀u ∈ Xh ‖Pu‖− 4 supv∈Xh

|ah(u, v)|

|v|. (154)

Proof. We first evaluate the norm of the restriction operator from a tetrahedronT ∈ Th to an edge e in norms H1+α(T )→ L2(e).

We let T be a reference tetrahedron of diameter 1. We suppose that thescaling x 7→ hx, maps the tetrahedron T to T and the edge e to e. For ak-multilinear form u on T , its pullback u to T satisfies, for x ∈ T :

u(x) = hku(hx). (155)

With this definition, we remark that ∇ commutes with the pullback by scalingmaps. Specializing to the case k = 2 we have the scaling estimates:

‖u‖L2(T ) = h−1/2‖u(x)‖L2(T ), (156)

‖∇u‖L2(T ) = h−3/2‖∇u(x)‖L2(T ), (157)

|u|Hα(T ) = h−3/2−α|u(x)|Hα(T ). (158)

From this we deduce:

|∫

etteute| = h−1|

∫

etteute| (159)

4 h−1‖u‖H1+α(T ) (160)

4 h−1/2‖u‖H1+α(T ). (161)

Therefore, summing over all tetrahedrons:(

∑

e

(∫

e tteute)

2)1/2

4 h−1/2‖u‖H1+α(S). (162)

By duality we get, for any family of reals ve ∈ R attached to the edges e:

‖∑

e

vetetteδe‖H−1−α(S) 4 h−1/2(

∑

e

v2e)1/2. (163)

By scaling, for such a family (ve) we also have:

‖∑

e

veρe‖L2(S) ≈ h−1/2(

∑

e

v2e)1/2. (164)

Pick now u ∈ Xh and define the family (ve) by:

curlt curlu =∑

e

vetetteδe. (165)

We define v ∈ Xh by:

v =∑

e

veρe, (166)

and can now write:

|〈curlt curlu, v〉|

|v|=|∑

e v2e

∫

etteρete|

|v|(167)

< h−1/2(∑

e

v2e)1/2 (168)

< ‖ curlt curlu‖H−1−α(S). (169)

This completes the proof.

21

Corollary 4.6. There is a lower bound uniform in h:

infu∈Vh

supv∈Vh

|ah(u, v)|

|u|‖v‖−= inf

u∈Vh

supv∈Vh

|a(u, v)|

‖u‖−|v|< 1. (170)

Proof. To obtain the uniform lower bound for the second term remark first thatProposition 4.1 yields for uh ∈ Vh:

‖u‖− ≈ ‖Pu‖−. (171)

Then apply Proposition 4.5.The equality reflects that a map and its adjoint have the same norm.

Therefore the abstract setting can be applied to prove (on toruses):

Theorem 4.7. Linearized Regge calculus yields a convergent method to approx-imate the eigenpairs of the Saint-Venant operator.

Acknowledgments

I thank Ragnar Winther for stimulating discussions and Douglas N. Arnold andRagnar Winther for generously sharing their notes on the linearized Einsteinequations with me. In particular this is where I learned of the relationship ofthese equations to elasticity.

This work, conducted as part of the award “Numerical analysis and simu-lations of geometric wave equations” made under the European Heads of Re-search Councils and European Science Foundation EURYI (European YoungInvestigator) Awards scheme, was supported by funds from the ParticipatingOrganizations of EURYI and the EC Sixth Framework Program.

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